-
Class7:Geodesics
Inthisclasswewilldiscusstheequationofageodesicinacurvedspace,howparticlesand
lightraysmoveinacurvedspace-time,andhowthismotionconnectstoNewton’sLaws
-
Class7:Geodesics
Attheendofthissessionyoushouldbeableto…
• …
recognizethegeodesicequation,whichconnectsthemotionofparticlestothespace-timemetric,usingChristoffelsymbols
• … calculatetheChristoffel
symbolsinsomesimplecases,suchasin2Dspacesortheweak-fieldlimit
• …
recognizethatdifferentparameterscanbeusedtodescribetheworldlineofparticlesandlightrays
• …
understandhowGRconnectstoNewton’sLawsforweakfields,andhowthetimeco-ordinatebehavesinthislimit
-
Geodesics
•
AfundamentalquestionforGeneralRelativitytoansweris,howdoesanobjectmoveinagravitationalfield?
• Howcanwefindanobject’sworldline𝑥"(𝜏)
intermsofitspropertime𝜏,whenfreelyfallingintheEarth’sframe𝑥"?
https://mathspig.wordpress.com/2014/01/23/2-one-rule-aerial-skiers-cannot-break/
-
Geodesics
•
A“straightlineonacurvedsurface”iscalledageodesic,whichminimizesthedistancebetween2points
• e.g.,geodesicsonasphericalsurfaceare“greatcircles”
•
InGR,objectstravelonageodesicincurvedspace-time,whichextremizes
thepropertimebetween2points
-
Geodesics
• Thesamemathematicshencedescribesboth
thegeometryofcurvedspacesand thegeometryofspace-time
• ThismathswasEinstein’sbiggestchallengeindevelopingGR!
https://www.pinterest.com.au/pin/407083253789581007/
-
Geodesicequation
• Objectsmoveonapathwhichextremizes thepropertime
• Since𝑑𝑠( =
−𝑐(𝑑𝜏(,thepropertimealongapathisgivenintermsofthemetricby,
- ∫ −𝑑𝑠(��
� =,- ∫ −𝑔"1𝑑𝑥
"𝑑𝑥1���
• Wecanmathematicallyfindthepath𝑥"(𝜏)
thatextremizesthevalueofthisintegral.Theresultisthegeodesicequation:
𝑑(𝑥"
𝑑𝜏( + Γ56" 𝑑𝑥5
𝑑𝜏𝑑𝑥6
𝑑𝜏 = 0
Γ56" =
12𝑔
"1 𝜕6𝑔15 + 𝜕5𝑔61 − 𝜕1𝑔56
-
Geodesicequation
• What’sthephysicalmeaningof;
;?<+ Γ56
" ;=@
;?;=A
;?= 0?
•
It’stheequationofmotionoffreely-fallingparticlesinacurvedspace-time.Particlestravellingonageodesicfeelnoforces
https://www.quantum-bits.org/?p=963
-
Geodesicequation
• What’sthephysicalmeaningof;
;?<+ Γ56
" ;=@
;?;=A
;?= 0?
• Theindices𝜅 and𝜆
aresummedover,hencethisrelationrepresents4differentialequations,whichcanbesolvedtodeterminethepathofaparticlethroughspace-time,𝑥"(𝜏)
• IfΓ56" = 0,thentheparticlehaszeroacceleration;
;?<= 0
andismovinguniformlyinaninertialframe– so𝜞 = 𝟎
intheabsenceofgravity
• Hence𝜞𝜿𝝀𝝁 representsa“force”duetogravity,whichis
curvingthepathoftheparticlethroughspace-time
-
Christoffel symbols
• ThevaluesofΓ56" aredeterminedbythespace-timemetric
𝑔"1,asΓ56" = ,
(𝑔"1 𝜕6𝑔15 + 𝜕5𝑔61 − 𝜕1𝑔56
• ThisobjectΓ56" issoimportantthatithasaname– the
Christoffel symbols
• Whatis𝒈𝝁𝝂?If𝑔"1 iswrittenasamatrix,then𝑔"1
istheinverseofthematrix
• 𝜕" =,-KKL, KK=, K;N, K;O
isdifferentiatingthemetricelements
• Theindex𝜈 issummedover
-
Christoffel symbols
• Theproblemisthat,sinceeachindexcantakeon4values,Γ56"
consistsof4×4×4 = 64 differentfunctionsingeneral!
• However,itiseasierforthespecialcaseswe’llconsider
https://www.edvardmunch.org/the-scream.jsp
-
Christoffel symbols
• Space-timecurvaturetellsmatterhowtomove
Metric𝑔"1
ChristoffelsymbolsΓ56
"Geodesicequations
Worldline𝑥"(𝜏)
-
Christoffel symbols
• Thereisacalculational
trickwhichcanmakeiteasiertodeterminetheChristoffel
symbolsinsomecases…
• Thegeodesicequationis: ;
;?<+ Γ56
" ;=@
;?;=A
;?= 0
•
Amathematicallyequivalentversionofthegeodesicequationis:𝑔"1
;
-
Motioninaweakfield
•
Animportantspecialcaseisaparticlemovingslowlyinaweak,staticgravitationalfield
• Inthiscase,thespace-timemetriccanbewritteninthe
form𝑔"1(𝑥U) = 𝜂"1 + ℎ"1(𝑥U),where𝜂"1 =−1 00 1
0 00 0
0 00 0
1 00 1
is
theMinkowski metric,and ℎ"1 ≪ 1
isasmallperturbationwhichdependsonlyonspatialco-ordinates𝑥U,nottime
• Foraslow-movingparticle,;=Y
;?≪ ;=
Z
;?where𝑖 = 1,2,3
-
Motioninaweakfield
• Usingtheseapproximations,thegeodesicequationsimplythat;
-
Geodesicsforlightrays
• Foralightray,thereisasubtletywhichwrecksourpreviousderivation–
𝑑𝑠 = 𝑑𝜏 = 0.Wecannotdescribethepathasafunctionof𝜏,since𝜏 = 0
always!(it’sa“nullgeodesic”)
•
Weneedtoparameterizetheworldlinebyadifferentco-ordinatecalledthe“affineparameter”,𝑥"(𝑝)
• Weendupwiththesameequation,;
;e<+ Γ56
" ;=@
;e;=A
;e= 0
-
Geodesicsforlightrays
• It’susefultoconsiderthewavevector 𝑘" = ;=>
;e,intermsof
which;g>
;e+ Γ56
" 𝑘5𝑘6 = 0
• Inphysicalterms,thewavevector 𝑘"
givesthefrequency(𝑘h)anddirectionofmotion(𝑘U)ofthelightray
• Intheabsenceofagravitationalfield,Γ = 0,hence;g>
;e=
0,hence𝑘" = constant
•
Thegravitationalfield“bends”thelightrayaccordingtoΓ,changingitsdirectionoftravel
